- [Instructor] In this chapter,
you will explore numerical methods.
These methods are great to use
when traditional techniques to solve differential equations
are inadequate.
Let's begin by understanding what numerical methods are.
So far, you have explored various techniques
to solve differential equations,
and these techniques focus on finding an exact solution.
But what do you do if these techniques
are extremely difficult or impossible to use?
Instead of finding an exact solution,
you can simply find an approximation
utilizing numerical methods.
Numerical methods are computational techniques
utilized in mathematics to find approximate solutions.
These methods are extremely helpful
when you are unable to find a solution
through traditional analytical techniques
like the ones explored prior in this course.
Numerical methods often heavily rely
on calculators and computers to calculate their results.
It is common to have software packages built
for numerical methods
in fields such as physics and engineering,
where many key problems need to be solved
with these approximations.
I will not show you how to code with these techniques
in this course,
but I recommend exploring it more on your own.
Numerical methods are great to use
with more complex differential equations,
especially when they are non-linear or have a high order.
They can help solve both initial value problems
and boundary value problems.
They are also used in various practical applications
like the ones mentioned earlier,
such as physics, engineering, and biology.
The key thing to remember
is that numerical methods are an approximation.
This means the result you'll get will not be 100% accurate.
There will always be some certain degree of error,
including when solving the same problem
using different devices
such as a computer versus a calculator
or two different computers.
Much of the time these errors are small,
but be careful since they can quickly increase
depending on the complexity of the equation being solved
and the technique being used.
You'll also want to make sure
you account for round off and truncation errors
when calculating with these numerical methods.
There are a few other considerations to keep in mind
when working with numerical methods.
First, note that these techniques
tend to be computationally heavy,
meaning that there's sometimes a trade-off
between accuracy, step size, and computational cost.
Some of these methods like Euler's method
can become unstable,
so you may need to use very small step sizes
in order to manage the stability.
There are many numerical approximation methods
out there to explore.
In this course, you'll focus on learning Euler's method,
Improved Euler's method, Runge-Kutta method,
stability and convergence,
and various practical applications
that will be explored in the last video of this chapter.
Numerical methods can be extremely useful
for finding approximate solutions
for complex differential equations.
The goal of this chapter
is to help you understand these methods
and how they operate,
but note that you can expand upon using these methods
with coding and various software packages.
Let's begin by exploring Euler's Method.